First, we need to simplify the expression inside the parentheses:
14.3 - 9 5/6

To subtract mixed numbers, we need to convert 9 5/6 to an improper fraction.
9 * 6 = 54
54 + 5 = 59
So, 9 5/6 is equivalent to 59/6.

Now, subtract 59/6 from 14.3:
14.3 - 59/6

To subtract a decimal from a fraction, we need to convert the decimal to a fraction with a denominator of 6:
14.3 = 143/10

Now we can subtract the fractions:
143/10 - 59/6

To subtract fractions, we need to find a common denominator, which in this case is 30:
(143/10)3/3 = 429/30
(59/6)5/5 = 295/30

Now, subtract the fractions:
429/30 - 295/30 = 134/30 = 67/15

So, 14.3 - 9 5/6 = 67/15

Now, we can continue simplifying the expression:
9/26 67/15 (-1/2) + 14

To multiply fractions, multiply the numerators together and the denominators together:
(9 67 -1) / (26 15 2) + 14

-603 / 780 + 14

Now, to add -603/780 to 14, convert 14 to a fraction with a denominator of 780:
14 = 10920/780

-603/780 + 10920/780 = 10317/780

Therefore, the simplified expression is: 10317/780.